Multifield, Multifrequency Bosonic Stars and a Stabilization Mechanism

Scalar bosonic stars (BSs) stand out as a multipurpose model of exotic compact objects. We enlarge the landscape of such (asymptotically flat, stationary, everywhere regular) objects by considering multiple fields (possibly) with different frequencies. This allows for new morphologies and a stabilization mechanism for different sorts of unstable BSs. First, any odd number of complex fields, yields a continuous family of BSs departing from the spherical, equal frequency, $\ell$-BSs. As the simplest illustration, we construct the $\ell =1$ BSs family, that includes several single-frequency solutions, including even parity (such as spinning BSs and a toroidal, static BS) and odd parity (a dipole BS) limits. Second, these limiting solutions are dynamically unstable, but can be stabilized by a hybrid-$\ell$ construction: adding a sufficiently large fundamental $\ell =0$ BS of another field, with a different frequency. Evidence for this dynamical robustness is obtained by nonlinear numerical simulations of the corresponding Einstein-(complex, massive) Klein-Gordon system, both in formation and evolution scenarios, and a suggestive correlation between stability and energy distribution is observed. Similarities and differences with vector BSs are anticipated.

Figure 1

$\ell =1$ BSs family.

Figure 2

ADM mass vs frequency for some of the single-frequency limits of the $\ell =1$ BSs family. (Inset) Fraction of the total mass in ${\mathrm{SBS}}_{-1}+{\mathrm{SBS}}_{+1}$ and in ${\mathrm{DBS}}_0$ along the static sequence $1+2$ in Fig. 1 for $w_2=0.94\mu$ (and $w_1/\mu =w_3/\mu$ in the $x$ axis).

Figure 3

Time evolutions of the energy density for five static BSs in the $\ell =1$ family. Each column shows six sequential snapshots of the $xz$ ($y=0$) plane (time running from top to bottom). From left to right the values of $w_1=w_3$ of the five models are: 0.955 (${\mathrm{DBS}}_0$), 0.945, 0.940 ($\ell -\mathrm{BS}$), 0.935, 0.933 (${\mathrm{SBS}}_{-1}+{\mathrm{SBS}}_{+1}$).

Figure 4

Time evolution (top to bottom) of two saturns: (left and middle left) ${\omega }_1/\mu =0.967$; (middle right and right) ${\omega }_1/\mu =0.974$. For each case the energy density and the real part of ${\mathrm{\Phi }}^{(3)}$ on the $xy$ plane are shown. The first (second) saturn is close to ${\mathrm{MBS}}_0$ (${\mathrm{SBS}}_{+1}$) and is stable (unstable).

Figure 5

Instability timescale (black dots and lines) and maximum of the total energy density (red lines) for the sequence of saturns (main panel) and pods (inset) described in the text. The blue points and line are thresholds when no instability is seen. The $z$ coordinate in the inset is $r$ for $\theta =0,\pi$. $T_{\text{instability}}$ refers to the time when we observe that the solution begins to clearly deviate from axisymmetry for saturns and from equatorial symmetry for pods.